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The Fundamental Theorem of Arithmetic
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Welcome class! Today, we are diving into the Fundamental Theorem of Arithmetic. Can anyone tell me what it states?
Is it something about prime factorization?
Exactly! The theorem states that every composite number can be expressed as a product of primes in a unique way. For example, how would you express the number 12?
Oh, that's 2 Γ 2 Γ 3!
Right! And if we rearrange it, it still gives you the same product. That's the uniqueness part. Can you think of another composite number?
What about 30? Itβs 2 Γ 3 Γ 5!
Perfect! Now, let's learn a memory aid: remember 'Prime is sublime' to recall that prime numbers play a special role in the foundation of integers.
I like that! It makes it easy to remember!
Great! So, if I asked you how many unique ways can 30 be expressed in terms of primes, what would you say?
Just one way, using 2, 3, and 5.
Exactly! Now, letβs summarize: The Fundamental Theorem shows that composites have unique prime factorization. Remember that!
Irrational Numbers
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Now that we've covered prime factorization, letβs talk about irrational numbers. Can someone remind me what an irrational number is?
It's a number that can't be expressed as a fraction of two integers.
Correct! Our goal is to prove that some integers, like 2 and 3, are irrational. Have any of you thought about how to prove this?
I think we can use the prime factorization!
Absolutely! Letβs assume 2 is rational and can be expressed as a fraction a/b, where a and b are coprime. What happens if we square this?
We get 2bΒ² = aΒ². So, doesn't that mean aΒ² is even?
Correct! And if aΒ² is even, what can we say about a?
A must also be even, and thus share a common factor with b, which contradicts our assumption.
Exactly! Thatβs how we show 2 is irrational. We can summarize this with: 'If square is even, base is even'.
HCF and LCM Workaround
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Now, let's simplify HCF and LCM using prime factorization. How do these two relate mathematically?
For HCF, we take the lowest powers of common primes, right?
Exactly! And for LCM?
We take the highest powers of each prime.
Good! Let's practice. If we have 30 and 42, what are the HCF and LCM?
The prime factorizations show that HCF = 6 and LCM = 210!
Very well done! Remember: 'HCF is Low, LCM is High', that can help you remember the prime power rules.
That's clever! I'll use that!
Introduction & Overview
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Quick Overview
Standard
In this section, we encapsulate the Fundamental Theorem of Arithmetic, which states that every composite number can be uniquely expressed as a product of prime factors, and explore how this theorem contributes to proving the irrationality of numbers like 2 and 3. These insights establish the foundational understanding for working with real numbers.
Detailed
Summary of Section 1.4
In this chapter, you have studied the following points:
1. The Fundamental Theorem of Arithmetic: This theorem establishes that every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, aside from the order of the prime factors.
2. Divisibility Relationships of Primes: If a prime number p divides the square of a positive integer aΒ², then p must also divide the integer a.
3. Proving Irrationality: You explored proofs that the numbers 2 and 3 are irrational, demonstrating how primes factor into the real number line.
4. HCF and LCM Relationships: Understanding of how HCF and LCM relate to prime factorization.
These key points emphasize the significance of prime numbers in determining the properties of integers and the structure of the number system.
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Fundamental Theorem of Arithmetic
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- The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
Detailed Explanation
The Fundamental Theorem of Arithmetic states that any composite number can be broken down into its prime factors uniquely, meaning there is only one way to express a composite number as a product of prime numbers. The order of the factors does not matter; for example, the number 12 can be factored as 2 Γ 2 Γ 3, or 2 Γ 3 Γ 2, or 3 Γ 2 Γ 2. All of these represent the same prime factorization.
Examples & Analogies
Think of a recipe for a cake. You can make a cake with certain ingredients in different sequences, but the ingredients themselves define what type of cake it is. Similarly, the prime factors define a composite number, and their unique combination (regardless of order) defines that number.
Divisibility of Primes
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- If p is a prime and p divides a2, then p divides a, where a is a positive integer.
Detailed Explanation
This statement tells us that if a prime number p divides the square of a positive integer a (represented as aΒ²), then it must also divide the integer a itself. For example, if we take p = 2 and a = 4, we see that 2 divides 4 (because 4 = 2 Γ 2), and it also divides 4's square, which is 16 (because 16 = 2 Γ 8).
Examples & Analogies
Imagine a group of students that can each hold a certain number of books. If there are 16 books and the number of books each student can hold is 4 (a group size), then we can say that this group configuration allows for evenly distributing books. If there's a larger number of books (such as 16 books), it just means that each student is still capable of holding their allotted amount.
Irrationality of Numbers
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- To prove that 2, 3 are irrationals.
Detailed Explanation
This point indicates that the numbers 2 and 3 cannot be expressed as fractions of two integers (in simplest form). This was shown through proofs that involved assuming they could be expressed as rational numbers and arriving at contradictions. For instance, assuming that β2 is rational leads to the conclusion that both integers in the fraction must have 2 as a common factor, which contradicts our assumption of them being coprime.
Examples & Analogies
Consider trying to measure the diagonal of a square with a side length of 1. If you attempt to express this length in terms of fractions (like 1/2, 1/3, etc.), you'll find that you cannot find exact fractions that equal the diagonal length. This is much like trying to find a fold in an origami that doesnβt exist, illustrating the concept of irrational numbers.
Relationships Between HCF and LCM
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A N R OTETOTHE EADER You have seen that : HCF (p, q, r) Γ LCM (p, q, r) οΉ p Γ q Γ r, where p, q, r are positive integers.
Detailed Explanation
This note highlights a common misconception regarding the product of the highest common factor (HCF) and lowest common multiple (LCM) of numbers p, q, and r. While it is true that the product of two numbers equals the product of their HCF and LCM, it does not hold when it involves three or more numbers. This distinction is crucial for understanding the relationships and calculations involving multiple integers.
Examples & Analogies
Think about three different pizza sizes. If you know how much topping to share among three sizes (small, medium, and large), you can calculate how much to put on each, but adding their individual toppings together wonβt necessarily equal what you'd expect for a combined total unless you re-calculate for the shared portions. Thus, just multiplying the individual totals doesnβt work when combined rationally.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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The Fundamental Theorem of Arithmetic: Every composite number has a unique prime factorization.
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Irrational Numbers: These cannot be expressed as fractions.
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HCF and LCM: Calculated using the prime factorization method.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example of prime factorization: 12 = 2^2 Γ 3.
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Example of proving irrationality: Show that β2 cannot be expressed as a fraction.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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Prime factorization is the game, unique for each number, that's its name!
π Fascinating Stories
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Once upon a time in Numberland, there was a party for all integers, but only the composites had unique prime invitations, showcasing their distinct nature in groups.
π§ Other Memory Gems
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Use βIrrational Isnβt Rationalβ to remember that irrational numbers cannot be written as fractions.
π― Super Acronyms
P.U.N.C.H. - Prime Uniqueness Neatly Counts as a Hint for prime factorization's uniqueness.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Composite Number
Definition:
A natural number greater than 1 that is not prime, meaning it has factors other than 1 and itself.
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Term: Prime Factorization
Definition:
The expression of a composite number as a product of its prime factors.
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Term: Irrational Number
Definition:
A number that cannot be expressed as a fraction of two integers.